Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds

• 成果类型: Article
• 署名作者: Cavalletti, Fabio; Mondino, Andrea
• 署名单位: University of Pavia; Swiss Federal Institutes of Technology Domain; ETH Zurich; University of Zurich
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0700-6
• 发表日期: 2017
• 页码: 803-849
• 关键词: dimension condition; li-yau; rcd-asterisk(k; alexandrov; geometry

摘要

We prove that if is a metric measure space with having (in a synthetic sense) Ricci curvature bounded from below by and dimension bounded above by , then the classic L,vy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by Milman for any , and upper diameter bounds) holds, i.e. the isoperimetric profile function of is bounded from below by the isoperimetric profile of the model space. Moreover, if equality is attained for some volume and K is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions. Finally we also establish the almost rigidity: if the equality is almost attained for some volume and K is strictly positive, then the space must be mGH close to a spherical suspension. To our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying Ricci curvature lower bounds. Examples of spaces fitting our assumptions include measured Gromov-Hausdorff limits of Riemannian manifolds satisfying Ricci curvature lower bounds, Alexandrov spaces with curvature bounded from below, Finsler manifolds endowed with a strongly convex norm and satisfying Ricci curvature lower bounds; the result seems new even in these celebrated classes of spaces.